In the prior art, seat areas and cone areas of rough ball point pen tips (tips without a tip ball inserted therein and prior to deformation to enclose the tip ball therein) were machined in succession by means of ordinary automatic machines with speed change disks in different successive stages of working. As a result, neither the eccentricity nor the burring was sufficiently well controlled. Subsequently, multi-part tools which could be maintained in such a way as to be mounted and fixed individually in a common clamping device were developed. This admittedly resolved the problem of eliminating burring, but concentricity accurate to a micrometer and the desired dimensions of the writing tips could only be achieved with the greatest difficulty, since there were no available high-speed high-precision spindles whose axis of rotation, from the stationary state to the maximum rotation speed, showed a deviation of less than 0.5 micrometers.